Understanding Decimal Notation

Date: 06/10/2002 at 00:03:37
From: Sharon Brown
Subject: Decimals

I have read your decimal page. I need to understand how the following 
problems can be worked out step by step. I need to see this process 
so it will stick in my head.

  9/10 = 0.9 

Why? How should it be worked out?

  12 15/100 = 12.15 

        0.7 = 7/10

       2.50 = 2 50/100

The explanation I was given is that you should be able to look at
these and work them out. I need to understand the process. Please
help!

Thanks

Date: 06/10/2002 at 11:58:03
From: Doctor Ian
Subject: Re: Decimals

Hi Sharon, 

When we write a number using decimal notation, it's really an 
implied sum.  That is, the notation '342.15' is really just a 
very compact way of writing the sum

  3*100 + 4*10 + 2*1 + 1*(1/10) + 5*(1/100)

That's what we _mean_ when we cram all those digits together in 
that order. 

Note that each term in the sum is a single digit multiplied by a 
power of 10:

      2       1       0       -1       -2
  3*10  + 4*10  + 2*10  + 1*10   + 5*10

If you're not sure how exponents work (especially zero and 
negative exponents), you might want to take a few minutes to look 
at this:

  Properties of Exponents
  http://mathforum.org/library/drmath/view/57293.html 

Now, the nice thing about the fractional part is that we can 
change between powers of 10 just by adding and taking away zeros, 
so 

   1    5         10    5         15
  -- + ---   =   --- + ---   =   ---
  10   100       100   100       100

So when you want to convert between decimals and fractions, all 
you have to do is find the power of 10 corresponding to the last 
digit, and then treat the others as an integer, e.g., 

                       1
   2.1       ->   2 + ---
                        1
                      10       <-- 10

                      4456
  12.4456    ->  12 + ----
                         4
                       10      <-- 10000

The shortcut is to count the number of decimal places, and put 
the same number of zeros after the 1 in the denominator:

                        56
  3.0056     ->   3 + -----
                      10000
     ^                  ^
     |                  |
   4 decimal         4 zeros
   places

But instead of trying to memorize the shortcut, you're better off 
making sure that you understand why it works, so that if you ever 
forget it, you can figure it out again on your own.  (In general, 
the best way to find out whether you understand something is to 
try to explain it to someone else who doesn't understand it.) 

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/11/2002 at 22:33:52
From: Sharon Brown
Subject: Thank you (Decimals)

Ian,

Thanks again for your help.